# Algorithms for Tverberg's theorem via centerpoint theorems

@article{Rolnick2016AlgorithmsFT, title={Algorithms for Tverberg's theorem via centerpoint theorems}, author={David Rolnick and Pablo Sober{\'o}n}, journal={arXiv: Computational Geometry}, year={2016} }

We obtain algorithms for computing Tverberg partitions based on centerpoint approximations. This applies to a wide range of convexity spaces, from the classic Euclidean setting to geodetic convexity in graphs. In the Euclidean setting, we present probabilistic algorithms which are weakly polynomial in the number of points and the dimension. For geodetic convexity in graphs, we obtain deterministic algorithms for cactus graphs and show that the general problem of finding the Radon number is NP…

## 6 Citations

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

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- 2019

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their…

Robust Tverberg and Colourful Carathéodory Results via Random Choice

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Borders are given for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.

The Crossing Tverberg Theorem

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A strengthening of Tverberg's theorem is proved that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections.

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This survey presents an overview of the advances around Tverberg’s theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg’s…

Journey to the Center of the Point Set

- MathematicsACM Trans. Algorithms
- 2021

An improved algorithm that can compute centerpoints with quality arbitrarily close to 1/d2 and runs in randomized time Õ(d7) randomized time is presented, the first refinement of the algorithm by Clarkson et al. in over 20 years.

Improved Approximation Algorithms for Tverberg Partitions

- Computer ScienceESA
- 2021

This work provides the first strongly polynomial (in both $n$ and $d$) approximation algorithm for finding a Tverberg point, and provides several new approximation algorithms for this problem, which improve either the running time or quality of approximation.

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